PDE: uf = 2ихх, 0 < х < 1, t> 0 0 0 ВС: и (0, t) — —1, иx(1,t) — 1 3πχ IC: и (х, 0) — х + sin - 1 2

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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need some help with partial diff equations.

PDE: uf =
2ихх, 0 < х < 1, t> 0
0 <x< 1, t > 0
ВС: и (0, t) —
—1, иx(1,t) — 1
3πχ
IC: и (х, 0) — х + sin
- 1
2
Transcribed Image Text:PDE: uf = 2ихх, 0 < х < 1, t> 0 0 <x< 1, t > 0 ВС: и (0, t) — —1, иx(1,t) — 1 3πχ IC: и (х, 0) — х + sin - 1 2
Expert Solution
Step 1

Claim: solution for the heat equation:

Advanced Math homework question answer, step 1, image 1

Let us use separation of variables. Let u(x,t)=X(x)T(t).

Then 4ut=uxx becomes X(x)T’(t)=X’’(x)T(t).

We divide both sides by X(x)T(t) to obtain:

Advanced Math homework question answer, step 1, image 2

where λ is a constant.

The boundary conditions are:

Advanced Math homework question answer, step 1, image 3

Integrating both sides

Advanced Math homework question answer, step 1, image 4

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